RAMIFICATION IN NUMBER FIELDS GENERATED BY x^4+p x^2+p WITH p \neq 4+n^2 A RATIONAL PRIME

If is the splitting field of the polynomial and is a rational prime, then we have , where is the dihedral group of order 8 . We calculate the discriminant of without knowing any integral basis of ; also, we obtain the explicit ramification of any rational prime such that . For this, we first study the ramification in the intermediate quartic field . We give generators of each ramified rational prime. In some cases, we can make use of Dedekind's Theorem; in other cases, we use the relative extension theory.